Extending the momentum-resolved study of vacuum-induced coherence in velocity-selective coherent population trapping of moving atoms to other quantum-optical systems

In 2008, Yang, Xie, Zhang, and Wang derived a momentum-resolved master equation for a Lambda-type atom undergoing velocity-selective coherent population trapping (VSCPT) while subject to vacuum-induced coherence (VIC) — the interference between two spontaneous-emission channels whose dipole moments are nonorthogonal and whose transition frequencies are nearly degenerate. They found that VIC generates off-diagonal momentum ('external') coherences that partially cancel the spontaneous-emission feeding of the dark trapping state, weakening trapping as the dipole angle between the two decay channels decreases. Noting that much less effort had been devoted to the atomic center-of-mass motion controlled by light, they closed by proposing a single future direction: extending this momentum-resolved study into other quantum-optical systems where the decoherence action of the quantum interference between different transition pathways to desired optical coherences must be considered. Vacuum-induced coherence has since been observed in cavity-QED, rubidium-atomic-beam, and x-ray nuclear settings, exploited in cold-atom continuum physics, and used in sub-Doppler laser-cooling proposals and analyses, but apart from the source group's own extension of this momentum-resolved analysis to a cold double-Lambda four-level atom (Ref. 4), no work reports VIC-generated momentum coherences in a velocity-selective trapping geometry or extends the momentum-resolved analysis to other quantum-optical systems with light-controlled atomic motion.

Background

Coherent population trapping (CPT) occurs in a Lambda-type three-level atom — two long-lived ground states $|g_1\rangle$, $|g_2\rangle$ coupled to a common excited state $|e_0\rangle$ by two coherent laser fields — when two-photon resonance locks the atom into a 'dark state', a coherent superposition of the ground states that decouples from the lasers because the two excitation amplitudes destructively interfere. When the two lasers counterpropagate and the atomic center-of-mass motion is quantized, the dark condition can only be met by atoms in a narrow momentum class, so repeated spontaneous-emission cycles pump atoms into those selected momenta: this is velocity-selective coherent population trapping (VSCPT), with which Aspect, Arimondo, Kaiser, Vansteenkiste, and Cohen-Tannoudji laser-cooled atoms below the one-photon recoil energy (Phys. Rev. Lett. 61, 826 (1988)). The natural unit of the momentum structure is the recoil frequency $\omega_r = \hbar k^2/2m$ for laser wave number $k$, and the standard theoretical tool is a family-basis master equation in which each momentum family $F(p)$ collects the states reached from momentum $p$ by absorbing or emitting laser photons.

Vacuum-induced coherence (VIC), also called spontaneously generated coherence (SGC), is a second interference mechanism that standard VSCPT theory omits. When two electric-dipole transitions share the same electromagnetic vacuum, are frequency-degenerate to within their linewidths, and have nonorthogonal dipole moments $\vec{d}_1$, $\vec{d}_2$, the two spontaneous-emission pathways interfere: the master equation acquires cross-damping terms proportional to $\sqrt{\gamma_1\gamma_2}\,\vec{d}_1\!\cdot\!\vec{d}_2$ (with $\gamma_{1,2}$ the individual decay rates), which can both suppress net decay and generate coherence between the two lower levels. The two conditions are jointly hard to satisfy in real atoms: transitions that are degenerate (e.g., Zeeman sublevels) typically have orthogonal polarizations, while transitions with nonorthogonal dipoles (e.g., distinct hyperfine levels) are split far beyond a linewidth. Consequently VIC was for decades a theoretical effect, explored in driven few-level systems and in proposals using anisotropic or engineered reservoirs — deliberately modified electromagnetic environments (cavities, nanostructures, or shaped fields) that reshape the vacuum modes to which the atom couples.

Yang, Xie, Zhang, and Wang (Phys. Rev. A 77, 063825 (2008), DOI: 10.1103/PhysRevA.77.063825) combined these two threads for the first time at the level of quantized atomic motion. For a Lambda atom driven by two laser fields and decaying from the common excited state to both ground states, they derived the momentum-resolved master equation including the VIC Liouvillian (within the Born-Markov approximation, which treats the vacuum as memoryless, and the rotating-wave approximation, which drops fast-oscillating terms) and solved it for the density-matrix elements $\rho_{ij}(p,p')$ in the momentum-family basis. Their key results are: (1) VIC source terms create off-diagonal external coherences $\rho_{ij}(p,p')$ with $p \neq p'$ that ordinary spontaneous emission never generates; (2) in the spatially frozen limit the dark-state feeding rate is proportional to $\gamma_1\Omega_1^2+\gamma_2\Omega_2^2-2\sqrt{\gamma_1\gamma_2}\,\Omega_1\Omega_2\cos\alpha$ (with $\Omega_{1,2}$ the Rabi frequencies and $\alpha$ the angle between the two dipole moments), which vanishes for symmetric parameters and parallel dipoles; but (3) when center-of-mass motion is retained the cancellation becomes only partial — it would require special equalities between momentum-diagonal and momentum-off-diagonal excited-state density-matrix elements — so VIC-induced external coherences counteract the spontaneous-emission feeding of the trapping state and monotonically degrade VSCPT efficiency as $\alpha$ decreases. They also showed that nondegenerate laser wave numbers add a detrimental recoil background to the momentum distribution. All numerical results were obtained for the symmetric, resonant parameter set $\Delta_1=\Delta_2=0$, $\Omega_1=\Omega_2$, $\gamma_1=\gamma_2$.

The paper closes by noting that, while the VIC effect on atomic trapping and related properties of optical systems had been widely discussed, much less effort had been devoted to the atomic center-of-mass motion controlled by light; its single stated future direction is that it may be worth extending the present study into other quantum-optical systems where the decoherence action of the quantum interference between different transition pathways to desired optical coherences must be considered. Two features of the paper bound this outlook: the nonorthogonal-dipole arrangement needed for VIC is treated within the paper itself (a possible arrangement is discussed in which the polarization of one field is perpendicular to the other transition dipole so that each field drives only one transition), and the VIC-induced external coherences are typically weak, so that in the absence of special ways to strengthen them the feeding effect of spontaneous emission plays a dominant role. Establishing how the VIC decoherence action on light-controlled atomic motion operates beyond the single Lambda VSCPT model — a question that touches both the foundations of radiative interference and the practical limits of subrecoil laser cooling (cooling to temperatures below the single-photon recoil energy) — is what remains open.

Problem Statement

Extend the momentum-resolved master-equation analysis of Yang, Xie, Zhang, and Wang (Phys. Rev. A 77, 063825 (2008), DOI: 10.1103/PhysRevA.77.063825) — in which vacuum-induced coherence generates external coherences $\rho_{ij}(p,p')$ with $p \neq p'$ that counteract the spontaneous-emission feeding of the VSCPT trapping state, the cancellation being only partial once the atomic center-of-mass motion is retained — to other quantum-optical systems in which light controls the atomic center-of-mass motion and the decoherence action of the quantum interference between different transition pathways to desired optical coherences must be considered. The symmetric, resonant Lambda VSCPT model solved in the source paper (within the Born-Markov and rotating-wave approximations, in the momentum-family basis $F(p)$ with recoil frequency $\omega_r = \hbar k^2/2m$) is the established starting point: the goal is to determine how the VIC-induced momentum coherences and their competition with spontaneous-emission feeding behave in further quantum-optical systems with quantized, light-controlled atomic motion. A single further system of this kind - one whose decay channels are nonorthogonal and nearly degenerate and which has not already been treated at this momentum-resolved level, that is, a system other than the source's Lambda three-level VSCPT model and the double-Lambda four-level atom of Ref. 4 - resolves the task when worked out to the same level of treatment as the source paper; a class-level result spanning several such systems is a strictly stronger outcome.

References

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